The Gromov-Witten theory of the Riemann sphere \(\mathbb {C}\mathbb {P}^1\) 
is known to be related to integrable hierarchies
of the Toda type. Such a link was first observed
by physicists employing random matrix models [1]}, [2]}.
This discovery was enhanced to a mathematical statement
called the Toda conjecture
[3]}, [4]}, [5]}.
The Toda conjecture was proved by several methods
[6]}, [7]}, [8]}, [9]}, [10]}
and generalized to \(\mathbb {C}\mathbb {P}^1\)  with orbifold points
[11]}, [12]}, [13]}, [14]}.
The relevant integrable hierarchies are
the 1D Toda hierarchy, the bigraded Toda hierarchy
[15]}, [16]} and a kind of
the Kac-Wakimoto hierarchies [17]}.
Another direction of generalization is the equivariant
Gromov-Witten theory of \(\mathbb {C}\mathbb {P}^1\) 
[7]}, [19]}, [20]}, [21]}.
The integrable hierarchies emerging therein are
the equivariant Toda hierarchy [22]}
and its bigraded version [20]}.
